Cantor Confusion
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From: mueckenh on 22 Apr 2007 05:32 On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > Each of these bunches contains an uncountable number > of paths. All you show is that there are a countable number > of bundles, each of which contains an uncountable number of paths. What about Cantor's list? Does it contain the representations of paths or of path bundles? If your answer is "paths", what is the difference to a path bunch with one element? > > Each of these paths can be separated from all other paths by > a set of nodes. There are uncountable many sets of nodes. > There are enough nodes to separate uncountable many paths. Your error lies in the fact, that one node can only once be used to separate two path bunches. Therefore the "sets of nodes" (which are nothing but paths) do not help to show the uncountability of paths. You are lacking some logic (quite a lot). You try to prove the uncountability of a set by the uncountability of that set. Regards, WM
From: William Hughes on 22 Apr 2007 07:08 On Apr 22, 5:32 am, mueck...(a)rz.fh-augsburg.de wrote: > On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > > > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > Each of these bunches contains an uncountable number > > of paths. All you show is that there are a countable number > > of bundles, each of which contains an uncountable number of paths. > > What about Cantor's list? An attempt to change the subject. Ignored. > Does it contain the representations of paths > or of path bundles? If your answer is "paths", what is the difference > to a path bunch with one element? > > > > > Each of these paths can be separated from all other paths by > > a set of nodes. There are uncountable many sets of nodes. > > There are enough nodes to separate uncountable many paths. > > Your error lies in the fact, that one node can only once be used to > separate two path bunches. Therefore the "sets of nodes" (which are > nothing but paths) do not help to show the uncountability of paths. I am not trying to show the "uncountablity of paths", just that there are enough nodes, so looking at the number of nodes does not settle the question in one direction of the other. > You are lacking some logic (quite a lot). You try to prove the > uncountability of a set by the uncountability of that set. > No, I show the possible uncountablity of entities capable of separating a single path from all other paths by showing that each such entity is a subset of a countable set. Since there are an uncountable number of such subsets we do not know if there are a countable or uncoutable number of such entities. Is the following statement true of false There are enough nodes to separate an uncountable number of paths. (Note the statement says nothing about whether or not the nodes do in fact separate an uncountable number of paths) - William Hughes
From: mueckenh on 22 Apr 2007 09:08 On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: > On Apr 22, 5:32 am, mueck...(a)rz.fh-augsburg.de wrote: > > > On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > > Each of these bunches contains an uncountable number > > > of paths. All you show is that there are a countable number > > > of bundles, each of which contains an uncountable number of paths. > > > What about Cantor's list? > > An attempt to change the subject. Ignored. Wrong. This is the important point. In Cantor's list numbers must be individuals. In the tree they cannot. > > > Does it contain the representations of paths > > or of path bundles? If your answer is "paths", what is the difference > > to a path bunch with one element? > > > > Each of these paths can be separated from all other paths by > > > a set of nodes. There are uncountable many sets of nodes. > > > There are enough nodes to separate uncountable many paths. > > > Your error lies in the fact, that one node can only once be used to > > separate two path bunches. Therefore the "sets of nodes" (which are > > nothing but paths) do not help to show the uncountability of paths. > > I am not trying to show the "uncountablity of paths", just that > there are enough nodes, so looking at the number of nodes > does not settle the question in one direction of the other. The number of path bunches is the same as the number of paths which can be distinguished by a node on a finite level. > > > You are lacking some logic (quite a lot). You try to prove the > > uncountability of a set by the uncountability of that set. > > No, I show the possible uncountablity of entities capable of > separating a single path from all other paths by showing > that each such entity is a subset of a countable set. That is a wrong claim. All the aleph0 nodes of the tree separate the following numbers of paths: 1 + aleph0 + 2-1-1+2-1-1+2-1-1... where the element 2-1-1 appears omega times. > > Is the following statement true of false > > There are enough nodes to separate an uncountable number > of paths. Definitively: False. Regards, WM
From: Virgil on 22 Apr 2007 12:48 In article <1177234322.313795.125580(a)l77g2000hsb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > > > Each of these bunches contains an uncountable number > > of paths. All you show is that there are a countable number > > of bundles, each of which contains an uncountable number of paths. > > What about Cantor's list? Does it contain the representations of paths > or of path bundles? Is a "path bundle" a set of paths? If so why not call it a set of paths? > If your answer is "paths", what is the difference > to a path bunch with one element? What is the difference between a "path bundle and a "path bunch"? > > > > Each of these paths can be separated from all other paths by > > a set of nodes. There are uncountable many sets of nodes. > > There are enough nodes to separate uncountable many paths. > > Your error lies in the fact, that one node can only once be used to > separate two path bunches. False! There are sets of paths which cannot be isolated from each other by any one node. > Therefore the "sets of nodes" (which are > nothing but paths) do not help to show the uncountability of paths. > You are lacking some logic (quite a lot). You try to prove the > uncountability of a set by the uncountability of that set. False! We succeed in proving uncountability of the set of paths of a CIBT by proving that the set of paths bijects with the uncountable power set of the set of levels of a CIBT. There are infinitely many levels in a CIBT. and for each subset of that infinite set of levels there is a unique path which ranches left at precisely those levels and branches right at all other levels. As this is a proof that WM cannot repudiate, he has to ignores it, or yield to logic.
From: Virgil on 22 Apr 2007 12:59
In article <1177247335.241896.62870(a)o5g2000hsb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 22 Apr., 13:08, William Hughes <wpihug...(a)hotmail.com> wrote: > > On Apr 22, 5:32 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > > On 21 Apr., 22:55, William Hughes <wpihug...(a)hotmail.com> wrote: > > > > > > On Apr 21, 4:33 pm, mueck...(a)rz.fh-augsburg.de wrote: > > > > > > Each of these bunches contains an uncountable number > > > > of paths. All you show is that there are a countable number > > > > of bundles, each of which contains an uncountable number of paths. > > > > > What about Cantor's list? > > > > An attempt to change the subject. Ignored. > > Wrong. This is the important point. In Cantor's list numbers must be > individuals. In the tree they cannot. Paths in a tree are as individual as any sort of binary sequences, as they are n more than binary sequences of left and right branchings. > > > > > Does it contain the representations of paths > > > or of path bundles? If your answer is "paths", what is the difference > > > to a path bunch with one element? > > > > > > Each of these paths can be separated from all other paths by > > > > a set of nodes. There are uncountable many sets of nodes. > > > > There are enough nodes to separate uncountable many paths. > > > > > Your error lies in the fact, that one node can only once be used to > > > separate two path bunches. Therefore the "sets of nodes" (which are > > > nothing but paths) do not help to show the uncountability of paths. > > > > I am not trying to show the "uncountablity of paths", just that > > there are enough nodes, so looking at the number of nodes > > does not settle the question in one direction of the other. > > The number of path bunches is the same as the number of paths which > can be distinguished by a node on a finite level. We are more interested in how many paths than in how many bunches. > > > > > You are lacking some logic (quite a lot). You try to prove the > > > uncountability of a set by the uncountability of that set. > > > > No, I show the possible uncountablity of entities capable of > > separating a single path from all other paths by showing > > that each such entity is a subset of a countable set. > > That is a wrong claim. It is a right claim that there are infinitely many levels in a CIBT, that corresond to the infinitely many natural number in N. It is a right claim that there is a unique and different path for every subset of the infinite set of levels, namely the path that branches left at each level in the subset and branches right at each level not in that subset. It is right to claim that the cardinality of the set of all paths equals the cardinality of the power set of N. Ergo, it is right to claim that the cardinality of the set of all paths is greater than that of N. > > > > > Is the following statement true of false > > > > There are enough nodes to separate an uncountable number > > of paths. > > Definitively: False. There are, nevertheless, as proved above, more paths than levels and more paths that members of N. |